Galileo's Paradox
in [Math]
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From: Tonico on 6 Dec 2006 06:09 Bob Kolker ha escrito: > David Marcus wrote: > > > > > > As Virgil pointed out, "no bijection to the naturals" is not a correct > > definition of "uncountable". > > Eh? > > See: http://en.wikipedia.org/wiki/Uncountable > > In mathematics, an uncountable or nondenumerable set is a set which is > not countable. Here, "countable" means countably infinite or finite, so > by definition, all uncountable sets are infinite. Explicitly, a set X is > uncountable if and only if there is an injection from the natural > numbers N to X, but no injection from X to N. > > From the wiki article on countability. > > Do you see in error in the wikipedia entry on uncountability > > > Bob Kolker ************************************************** Hi: I think you misunderstood something: we don't have a bijection from the set {x} to the naturals, and {x} is uncountable. So I think Virgil is right: no bijection to the naturals is NOT a correct definition of uncountable UNLESS we add the condition "...from an infinite set ...." Of course, perhaps you agree with this and I misunderstood something. Regards Tonio
From: Bob Kolker on 6 Dec 2006 06:10 David Marcus wrote: > > > Analogy: mind is software, brain is hardware. underline the word -analogy-. Mind, as conceived of by the classical philosophers, is a substance which differs from material substances. Res Cogitens vs. Res extensa. This is essentially Cartesian Dualism and is empirically unfounded. Another analogy. Brain is the instrument. Mind is the music. In any case mind appears to be an epiphenomenon of the brain. It is an effect of the physical activities of the brain. No brain, no mind. Mind is not a stand-alone object. That is why ten thousand years of humans slicing and dicing each other's bodies has never revealed a mind. Bob Kolker >
From: Tonico on 6 Dec 2006 06:10 Tonico ha escrito: > Bob Kolker ha escrito: > > > David Marcus wrote: > > > > > > > > > As Virgil pointed out, "no bijection to the naturals" is not a correct > > > definition of "uncountable". > > > > Eh? > > > > See: http://en.wikipedia.org/wiki/Uncountable > > > > In mathematics, an uncountable or nondenumerable set is a set which is > > not countable. Here, "countable" means countably infinite or finite, so > > by definition, all uncountable sets are infinite. Explicitly, a set X is > > uncountable if and only if there is an injection from the natural > > numbers N to X, but no injection from X to N. > > > > From the wiki article on countability. > > > > Do you see in error in the wikipedia entry on uncountability > > > > > > Bob Kolker > ************************************************** > Hi: > I think you misunderstood something: we don't have a bijection from the > set {x} to the naturals, and {x} is uncountable. > So I think Virgil is right: no bijection to the naturals is NOT a > correct definition of uncountable UNLESS we add the condition "...from > an infinite set ...." > Of course, perhaps you agree with this and I misunderstood something. > Regards > Tonio *********************************** Of course, I meant above "...and {x} is NOT uncountable" tonio
From: Bob Kolker on 6 Dec 2006 06:11 Virgil wrote:> > EB's arguments give us a plethora of examples of both unfoundedness and > the uselessness. That is why EB should be given the Zick Prize. Bob Kolker
From: Bob Kolker on 6 Dec 2006 06:18
Eckard Blumschein wrote: > > > A subset inside the reals is comparable to a piece of sugar within tea. > No. The elements of a piece of sugar are not tea. A is a subset of B if and only if every element in A is an element in B. Why do you make a simple concept more difficult with inept and inapt analogies, when a straightforward definition is at hand? Bob Kolker |